Junping Wang

Junping Wang, Xiu Ye

In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called {\em discrete weak gradients} which is expected to play important roles in numerical methods for partial differential equations. This article intends to provide a general framework for operating differential operators on functions with heterogeneous properties. As a demonstrative example, the discrete weak gradient operator is employed as a building block to approximate the solution of a model second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical approximation is called a weak Galerkin (WG) finite element solution. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete $H^1$ and $L^2$ norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.

Lin Mu, Junping Wang, Guowei Wei et al.

Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element.

Lin Mu, Junping Wang, Xiu Ye

This paper introduces a new weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This method, called WG-FEM, is designed by using a discrete weak gradient operator applied to discontinuous piecewise polynomials on finite element partitions of arbitrary polytopes with certain shape regularity. The paper explains how the numerical schemes are designed and why they provide reliable numerical approximations for the underlying partial differential equations. In particular, optimal order error estimates are established for the corresponding WG-FEM approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the WG-FEM. All the results are derived for finite element partitions with polytopes. Allowing the use of discontinuous approximating functions on arbitrary polytopal elements is a highly demanded feature for numerical algorithms in scientific computing.

Chunmei Wang, Junping Wang

In this paper, the authors devise a new discretization scheme for div-curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in the algorithm development: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is devised and analyzed. The novelty of the technique lies in the discretization of the divergence operator applied to vector fields with heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.

Chunmei Wang, Junping Wang

This paper presents a primal-dual weak Galerkin (PD-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. The method is based on a variational form where all the derivatives are applied to the test functions so that no regularity is necessary for the exact solution of the model equation. The numerical scheme is designed by using locally constructed weak second order partial derivatives and the weak gradient commonly used in the weak Galerkin context. Optimal order of convergence is derived for the resulting numerical solutions. Numerical results are reported to demonstrate the performance of the numerical scheme.

Chunmei Wang, Junping Wang

The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak $L^2$ topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.

Yujie Liu, Junping Wang

This article presents a simplified formulation for the weak Galerkin finite element method for the Stokes equation without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak Galerkin algorithm. The simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions. A particular application of the simplified weak Galerkin on rectangular partitions yields a new class of 5- and 7-point finite difference schemes for the Stokes equation. An explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements. Error estimates of optimal order are established for the simplified weak Galerkin finite element method in the H^1 and L^2 norms. Furthermore, a superconvergence of order O(h^{1.5}) is established on rectangular partitions for the velocity approximation in the H^1 norm at cell centers, and a similar superconvergence is derived for the pressure approximation in the L^2 norm at cell centers. Some numerical results are reported to confirm the convergence and superconvergence theory.

Long Chen, Junping Wang, Yanqiu Wang et al.

In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a "coarse" space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In this construction, the authors conveniently use the $H^1$ conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the $H^1$ conforming piecewise linear finite element space. The authors successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as a reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments are conducted to support the theoretical results.

Lin Mu, Junping Wang, Xiu Ye

A discrete divergence-free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in the reference [15]. Discrete divergence-free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence-free basis derived, the discrete divergence-free WG scheme can eliminate the pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence-free WG method.

Mu Lin, Junping Wang, Yanqiu Wang et al.

This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [17]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.

Lin Mu, Junping Wang, Xiu Ye et al.

A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimate. This refined interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.

Lin Mu, Junping Wang, Xiu Ye

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique.

Yujie Liu, Junping Wang, Qingsong Zou

This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each element. The numerical scheme is based on a constrained flux optimization approach where the constraint was given by local mass conservation equations and the flux error is minimized in a prescribed topology/metric. This new scheme provides numerical approximations for both the primal and the flux variables. It is shown that the numerical approximations for the primal and the flux variables are convergent with optimal order in some discrete Sobolev norms. Numerical experiments are conducted to confirm the convergence theory. Furthermore, the new scheme was employed in the computational simulation of a simplified two-phase flow problem in highly heterogeneous porous media. The numerical results illustrate an excellent performance of the method in scientific computing.

Yujie Liu, Junping Wang

In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations. The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method. A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions. Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.

Lin Mu, Junping Wang, Xiu Ye et al.

Weak Galerkin (WG) refers to general finite element methods for partial differential equations in which differential operators are approximated by weak forms through the usual integration by parts. In particular, WG methods allow the use of discontinuous finite element functions in the algorithm design. One of such examples was recently introduced by Wang and Ye for solving second order elliptic problems. The goal of this paper is to apply the WG method of Wang and Ye to the Helmholtz equation with high wave numbers. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Our numerical experiments indicate that weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

Wenhua Wu, Qi Wang, Guangming Wang et al.

Road surface reconstruction plays a vital role in autonomous driving systems, enabling road lane perception and high-precision mapping. Recently, neural implicit encoding has achieved remarkable results in scene representation, particularly in the realistic rendering of scene textures. However, it faces challenges in directly representing geometric information for large-scale scenes. To address this, we propose EMIE-MAP, a novel method for large-scale road surface reconstruction based on explicit mesh and implicit encoding. The road geometry is represented using explicit mesh, where each vertex stores implicit encoding representing the color and semantic information. To overcome the difficulty in optimizing road elevation, we introduce a trajectory-based elevation initialization and an elevation residual learning method based on Multi-Layer Perceptron (MLP). Additionally, by employing implicit encoding and multi-camera color MLPs decoding, we achieve separate modeling of scene physical properties and camera characteristics, allowing surround-view reconstruction compatible with different camera models. Our method achieves remarkable road surface reconstruction performance in a variety of real-world challenging scenarios.

Junping Wang, Bicheng Wang, Yibo Xuea et al.

Resilience non-equilibrium measurement, the ability to maintain fundamental functionality amidst failures and errors, is crucial for scientific management and engineering applications of industrial chain. The problem is particularly challenging when the number or types of multiple co-evolution of resilience (for example, randomly placed) are extremely chaos. Existing end-to-end deep learning ordinarily do not generalize well to unseen full-feld reconstruction of spatiotemporal co-evolution structure, and predict resilience of network topology, especially in multiple chaos data regimes typically seen in real-world applications. To address this challenge, here we propose industrial brain, a human-like autonomous cognitive decision-making and planning framework integrating higher-order activity-driven neuro network and CT-OODA symbolic reasoning to autonomous plan resilience directly from observational data of global variable. The industrial brain not only understands and model structure of node activity dynamics and network co-evolution topology without simplifying assumptions, and reveal the underlying laws hidden behind complex networks, but also enabling accurate resilience prediction, inference, and planning. Experimental results show that industrial brain significantly outperforms resilience prediction and planning methods, with an accurate improvement of up to 10.8\% over GoT and OlaGPT framework and 11.03\% over spectral dimension reduction. It also generalizes to unseen topologies and dynamics and maintains robust performance despite observational disturbances. Our findings suggest that industrial brain addresses an important gap in resilience prediction and planning for industrial chain.

Yujie Liu, Junping Wang

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection-diffusion-reaction problems obtained from the weak Galerkin finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the weak Galerkin involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin method has a reduced computational complexity over the usual weak Galerkin, and indeed provides a discretization scheme different from the weak Galerkin when the reaction term presents. An application of the simplified weak Galerkin on uniform rectangular partitions yields some $5$- and $7$-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the discrete maximum principle and the accuracy of the scheme, particularly the finite difference scheme.

Chunmei Wang, Junping Wang

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Optimal order error estimates are derived for the finite element approximations in a discrete $H^2$-norm, as well as the usual $H^1$- and $L^2$-norms. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains.